The Catalan giants want to sign competition for Jordi Alba at left-back next season, and Alonso has seemingly emerged as the club's leading candidate to fill the role.
According to Sport, Barcelona are now hopeful of signing the Spain international for a fee in the region of €10m (£8.6m), with his contract at Stamford Bridge due to expire next June.
Alonso recently admitted that he wants to return to Spain this summer but is "happy at Chelsea", having been a regular for the Blues during the 2021-22 campaign.
"On paper, I have a contract for one more year and that is where we're at today. I've already said that I would like to play in Spain again, but it doesn't depend entirely on me," he told reporters.
"I can't complain about anything at Chelsea, but the desire to play in Spain is there. It doesn't depend entirely on me and I'm happy at Chelsea, honestly."
Meanwhile, the Spaniard has now made 211 appearances for the Blues since his arrival from Fiorentina in 2016, contributing 29 goals and 23 assists in the process.
Alonso has won the Premier League, FA Cup, Champions League, Europa League, UEFA Super Cup and Club World Cup during a successful period at Stamford Bridge.
The defender has often spoken of his desire to return to Spain, though, having played in England and Italy since leaving Real Madrid in 2010.
Alonso came through the youth system at Los Blancos and was a regular for Real Madrid Castilla between 2008 and 2010, but he left the capital giants having made just one first-team appearance.
Barcelona have allegedly already agreed to sign Andreas Christensen following his release from Chelsea, although there has not yet been an official announcement confirming the move.
Barcelona will take on Olot, Inter Miami, Real Madrid, Juventus and New York Red Bulls in friendlies this summer ahead of their 2022-23 La Liga opener against Rayo Vallecano on August 13. body check tags ::